The formula (REF ) implies that our estimator \(\hat{f}_{n,V}^d\)  can be viewed as a least squares estimator over a
smoothness-constrained function class where smoothness is
measured in terms of the \(L_1\)  norms of mixed partial derivatives of orders up to 2.
It is well-known from approximation theory that \(L_p\) 
norm constraints on mixed partial derivatives are very advantageous
and allow one to overcome the curse of dimensionality to some extent
from the perspective of metric entropy, approximation, and
interpolation (see, e.g., [1]},
[2]}, and [3]}). These
results hint that our LASSO variant of the MARS method can be effective in
multiple dimensions, and we state formal rate results in Section
 demonstrating this.
